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Quarks have electric charges proportional to one third of the elementary electric charge, but both mesons and baryons have integer electric charge.

Is there some deep explanation from a more fundamental and generalizable conservation law that any state that forms an $SU(3)$ singlet must have an integer electric charge? Are there any common beyond-the-standard-models (BSM) where this does not hold?

I can show this for systems with just quarks & gluons, since quarks carry -1 electric charge quanta (1/3 of elementary charge) mod 3, antiquarks carry 1 electric charge quanta mod 3, and gluons carry 0 electric charge quanta mod 3 (same as a quark + antiquark pair), which implies that for any $SU(3)$ representation $D(p,q)$ composed of these, we must have that the electric charge mod 3 is equal to $(p-q)$ mod 3. So an integer Baryon number implies an integer electric charge (in terms of elementary charges). The same derivation also works for the U(1) weak hypercharge.

But this derivation just uses the explicit list of fundamental particle in the standard model with tables of their electric & color charges, and doesn't necessarily put it into any wider context. Is this just a coincidence in the standard model? Or is it a property of some GUTs as well? It feels like a really crazy coincidence that the SU(3) and the U(1)xSU(2) sectors are related to each other in this way.

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It may help noticing that the normalization of the $U(1)$ charge is arbitrary. The only meaningful information that one can have is the ratios of the charges of the particles in the model (2:(-1) for the case of the quarks). They can always be defined to be all integer numbers.

If you want to know why the charge of the color singlets is an integer number times the electron charge (which has apparently no relation with the color sector) then your question is more about an explanation for the ratios of charges between quarks and leptons. In that case, you may find useful these other two questions (the first one is mentioned in the comments):

Is there an explanation for the 3:2:1 ratio between the electron, up and down quark electric charges?

What's the deepest reason why QCD bound states have integer charge?

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