Is there an unit of color charge? I haven't found it, so I suppose that it doesn't exist, if this is right, why? Isn't it supposed that every measurable quantity can be expressed in terms of base quantities? What about flavour charge?
Answer
The colour charge of quantum chromodynamics is, as far as we can tell, not experimentally measurable, because of quark confinement. More specifically, quantum chromodynamical systems are always colour neutral. Quarks do have a color charge, but they are always observed in groups of two (color + anticolor) or three (red + green + blue) for which the total colour charge is zero (i.e. "white"). If you were to observe a lone quark then you'd be able to measure its charge and (up to a messier conventions hassle than with electric charges) its "sign", i.e. colour.
What you can measure, on the other hand, is the coupling between different colour charges. This has a direct equivalent in QED: electric charge carries only artificial units, and in natural units the relevant parameter is the coupling $$\alpha=\frac{e^2}{\hbar c}.$$ Similarly, there is a QCD coupling constant, which is complicated as it depends on distance $\leftrightarrow$ enegy. Just as $e^2$, it is dimensionless in natural units, which are the only ones you want to be working with anyway. (In other units, $e^2$ has units of $\text{energy}\times\text{length}$, but I would think this does not carry over to QCD.)
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