The Basic Forces and Messenger Particles — Question 6

Question 6

Gluon color degrees of freedom exclude the singlet combination

Color triplet and antitriplet indices give nine formal color-anticolor combinations. In SU(3), $3\otimes\bar3=8\oplus1$. Gauge fields lie in the traceless part.

1. Count independent gluon color states and identify the removed combination.

2. For diagonal coefficients a,b,c with a+b+c=0, count independent diagonal directions and add the off-diagonal ones.

Question 6 - Solution

Gluon color degrees of freedom exclude the singlet combination

(a) Method and calculation

$$\boxed{N_g=3^2-1=8}.$$$$\boxed{|{\mathrm{singlet}}\rangle=(|r\bar r\rangle+|g\bar g\rangle+|b\bar b\rangle)/\sqrt3}.$$

(b) Interpretation and further calculation

$$c=-a-b\Rightarrow\boxed{N_{\mathrm{diagonal}}=2}.$$$$\boxed{N_{\mathrm{off-diagonal}}=3(3-1)=6,\quad2+6=8}.$$

Physical check

Gluons are massless in the Standard Model, but confinement prevents a freely propagating macroscopic color force analogous to electrostatics.