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Gell-Mann's $SU(3)$ quark model is extremely successful at describing the bound states of the three light quarks $u,d,s$. The bound states fall neatly into the irreducible representations of $\mathfrak{su}(3)$. With the recent discovery of the doubly charmed baryon $\Xi_{cc}^{++}$ I have been thinking about how this "eightfold way" may be extended to include all six flavors. Is it as simple as extending the flavor symmetry group to $SU(6)$? I am a little rusty on my group theory, but if I remember correctly $SU(3)$ is not isomorphic to a subgroup of $SU(6)$. So how could this extension preserve the highly successful theory of $SU(3)$ for the lighter quarks?

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SU(3) is a subgroup of SU(6): its generators span the 3×3 block of the 6×6 generators of the latter.

Yes, the eightfold way trivially extends to a notional 35-way of all six flavors, which is, nevertheless, largely useless.

You may, of course, build classification tables of such hadrons in a 5-dimensional space (rank of SU(6)), but with little logical benefit. People did build 3d pictures of the first 4 quarks' SU(4) when charm was discovered,

charmed states

However, the whole point of the 8-fold way was that the lightest 3 quarks are lighter than the scale of QCD, ~200 MeV , which binds them together, so their masses could serve as small perturbations to a robust underlying pattern, explicitly broken (corrected) by small SU(3) violations.

For more flavors, the violations are, evidently, huge, and so your proposal might be a fool's errand.

Perversely, the 3 heaviest quarks, c,b,t, form a separate SU(3) of their own, as they are so heavy that you may transcribe across their differences, and treat the much lighter (scale-separated) QCD "glue" as invariant around them: "brown mud", in Bjorken's words, for example.

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