Quark-Gluon color relationship in pure QCD

Question

Consider pure QCD, flavor turned off. There are 3 quarks and 3 anti-quarks given by the fundamental IRR of SU(3) and its conjugate rep. We associate the three colors and anti-colors with the three vertices (weights) in the triangular weight diagrams of each of these two IRRs. (i.e. with the eigenvalue pairs of the Cartan subalgebra of su(3).) A quark "measured" (if that were possible) at any moment must be at one of these vertices = carrying one of those colors = having that weight.

The gluons must correspond to the adjoint IRR of SU(3) so are represented by the octet weight diagram of SU(3) (the regular hexagon + two weights at the origin.) Gauge theory requires gluons to carry both color and anti-color. (They are their own anti-particles, no anti-gluons, so must be able to interact with both quarks and anti-quarks, hence must carry both types of color.) Thus each vertex of the octet diagram must represent a quark-type (color,anti-color) pair or possibly a linear superposition thereof, for gluon-quark interaction to occur via color. But the octet weights (vertices) are ev's for the adjoint IRR, with no apparent direct relationship to those of the fundamental IRRs, other than indirectly perhaps via 3X3* = 8 + 1 or something similar.

It seems that there should be some sort of mathematical relationship (expression?) between the weights of the triplet diagrams and those of the octet diagram (i.e. between the eigenvalue sets) for us to be able to claim that physically the colors and anti-colors we are assigning to the gluons actually "connect" to (annihilate and create) those of the quarks and anti-quarks. (The triplet weight diagrams fit well inside the octet diagram by factors of 1.5 - 3.) The triplets and octet are separate eigenvalue sets of different IRRs that, a priori, don't seem to have any simple direct connection. Is there one?

Note: I am aware of the standard representation of the gluon colors as linear superpositions of (color,anti-color) pairs but this does not seem to me to motivate the above IRR evs connection, only decree that it exists. The derivation of that representation may be the answer.

Answer 1

There are three vectors in the quark representation, you can give them color names if you want, but your question seems to be about weights. From your question you already know this, but the weights of the quarks are just the eigenvalues under the two Cartan subalgebra generators ($T_3$ and $T_8$ in the usual convention).

So the three quarks are $(\pm \frac{1}{2},\frac{1}{2\sqrt{3}})$ and $(0,-\frac{1}{\sqrt{3}})$. The antiquark representation is just the negative of each component.

Now the weights of the gluon (octet) representation are $(\pm \frac{1}{2},\pm\frac{\sqrt{3}}{2}), (0,\pm1), $ and $(0,0)$. The last pair of zero eigenvalues is associated to the two gluons corresponding to the Cartan subalgebra, and it is also associated to the singlet representation.

Now if a gluon goes to a tensor product of quark and antiquark, this interaction respects the SU(3) symmetry so its eigenvalues under the Cartan generators must be the same before and after.

So for instance, you can see by adding the eigenvalues that the only allowed interaction for the $(\frac{1}{2},\frac{\sqrt{3}}{2})$ gluon going to quark $\otimes$ antiquark is $$(\frac{1}{2},\frac{\sqrt{3}}{2})\rightarrow( \frac{1}{2},\frac{1}{2\sqrt{3}})\otimes(0,+\frac{1}{\sqrt{3}})$$ in this sense it can be associated to a unique quark and antiquark, as can all six of the gluons not in the Cartan subalgebra, and thus you can give those six color names if you really want to do that for some reason.