Is color charge quantized?

Question

I was reading this stackexchange question, and found the answer to my question not totally answered. Clearly there is color and anti-color in analogy to electric charge, and color charge clearly cannot vary from color to anti-color. However can color (or anti-color) continuously vary between a red green and blue basis, or is it like wavelengths in atomic orbitals where in order to go from one color to another you have emit the exact amount of color charge required?

Answer 1

Naively, color can vary continuously between the colors according to a gauge transformation $\psi\mapsto \mathrm{e}^{\mathrm{i}\epsilon^a T^a}$ for some $\mathfrak{su}(2)$-valued object $\epsilon$, this is precisely the same as saying that a particle with electrical charge $e$ can vary continuously in phase according to $\psi\mapsto \mathrm{e}^{\mathrm{i}e\phi}\psi$.

However, there is a crucial difference: The $\mathrm{U}(1)$ symmetry of electromagnetism is Abelian, and so all transformations with constant $\phi$ are global symmetry transformations that have no gauge character, since the gauge field does not change under such transformations. In contrast, the $\mathrm{SU}(3)$ symmetry group is non-Abelian, and even constant $\epsilon$ change the gauge field, unless they commute with it. The set of elements of a non-Abelian group that commute with all others is called the center, and the center of $\mathrm{SU}(N)$ is the discrete group $\mathbb{Z}_n$.

So while electrically charged matter retains a continuous $\mathrm{U}(1)$ symmetry even after eliminating the gauge, color-charged matter retains only a discrete $\mathbb{Z}_3$ symmetry. That is, if you eliminate the gauge (which, in general, we cannot do: Gribov ambiguities prevent us even in principle from doing so globally, and even then, we will face a loss of covariance) you will end up with a particular set of red/blue/green particles that no longer can transform into each other. In this gauge-fixed world, you could think of color as a fixed property of each object, but this is not a useful intuitive picture to have. We describe the world through gauge theories precisely because the gauge-less description is not tractable.

However, that there is a discrete global $\mathbb{Z}_3$ symmetry is a valuable insight, as this is what is actually broken in the Higgs mechanism, as explained in this answer by Dominic Else.