Is the concept of bicolored gluons mathematically precise/meaningful? Please explain Each flavour of quark carries a colour quantum number: red, green or blue. I know what it means mathematically. But elementary textbooks (e.g, particle physics by Griffiths) also say that gluons are bicoloured. I don't understand this. Is there a rigorous mathematical meaning to this phrase? Can we understand what the phrase "bicoloured" means from the structure/symmetry of the QCD Lagrangian? I have read Yang-Mills theory but this thing is not clear to me.
 A: Quarks have color charge, but antiquarks have anticolor charges.  Baryons form color singlets by including three quarks with all three colors. But mesons form color singlets by having a quark and an antiquark with a color charge and its corresponding anticolor.  (Or better, a combination of the color-anticolor matchups with the appropriate symmetries under rotations in the SU(3) color space.)
Gluons likewise have a color and an anticolor. Think of a gluon as an operator which changes a quark’s color charge. A red quark can become a blue quark by emitting a red-antiblue gluon. The gluon in turn, can’t get far before being absorbed by some blue quark or by some antired antiquark.
Your favorite textbook on SU(3) matrix representations will explain why there are eight linearly independent gluons and help you construct their representations in color space.  (Short version: the ninth operator in the group is the identity.)
A: Gluons carry two color indices, the gluon field can be written as linear combination of Gell-Mann matrices: https://en.wikipedia.org/wiki/Color_charge#Quark_and_gluon_fields

The meaning is the following. Let $\psi_i$
represent the i-th component of a quark field (loosely called the i-th
color). The color of a gluon is similarly given by
$\mathbf {A}$ which corresponds to the particular
Gell-Mann matrix it is associated with. This matrix has indices i and
j. These are the color labels on the gluon.

A: This should be a comment, but I do not know how to put figures in comments.

Gluon interactions are often represented by a Feynman diagram. Note that the gluon generates a color change for the quarks. The gluons are in fact considered to be bi-colored, carrying a unit of color and a unit of anti-color as suggested in the diagram at right. The gluon exchange picture there converts a blue quark to a green one and vice versa.


italics mine.
