Let's have two quarks, which refers to representations of $r_{1}$ and $r_{2}$ of color symmetry group. They create bounded state which refers to the representation $r$.

There is a statement that measure and sign of interactions between them are determined by the value $$ \tag 1 \frac{1}{2}\left(C(r_{1}) + C(r_{2}) - C(r) \right)g^{2} $$ ($g$ is coupling constant).

I have two questions:

1) How to prove $(1)$?

2) How to calculate the values of Casimir operators if I have $3 \otimes \bar{3} = 8 \oplus 1$ channel? What I need to do: to calculate Casimir operators values for all of representations, and then to assume that $r_{1}$ refers to $3$, $r_{2}$ - to $\bar{3}$ and $r$ - to $8$ or to $1$? What are the differences between Casimir operators for $3$ (quarks) and for $\bar{3}$ (antiquarks)?

  • $\begingroup$ $(1)$ is not a statement that could have a truth value, so you can't prove it, it's just an expression. You'll have to write what quantity is supposed to be equal/proportional to this in order to have something to prove. For your second question, the Casimir operators of which representation do you want (I suppose of $\mathbf{1}$,$\mathbf{3}$,$\mathbf{8}$?)? $\endgroup$ – ACuriousMind Oct 28 '14 at 18:39
  • $\begingroup$ @ACuriousMind : yes, I'd like to know how to determine values of Casimir operators for a given representations. $\endgroup$ – user8817 Oct 28 '14 at 18:43
  • $\begingroup$ What's wrong with just evaluating the sum of the square of the generators in the given representations? (That's kinda the definition of the Casimirs, after all) $\endgroup$ – ACuriousMind Oct 28 '14 at 18:45
  • $\begingroup$ @AcuriousMind : I have edited the question. $\endgroup$ – user8817 Oct 28 '14 at 18:47