For any , it is a matrix constrained by the equation where Therefore, the dimension of is .
If we choose a basis of such that then where and Therefore, because .
Especially, if , then where . Now take , since , It’s also true in the case of .
When calculating Feynman diagrams in the usual Yang-Mills theory,
we use 3-point vertexs with coefficients and 4-point vertexs to build the gluon
amplitude, where is the structure constant of the (semi-simple) gauge group. It can be calculated
by the following formula in our basis. Therefore, for gauge field theory, in the calculation of tree level gluon
amplitude, the following formula is fundamental: Similarly, we can calculate that and so on. These expressions are both hold for and with the different
meanings of generators .
Therefore, the color structure of gauge field theory is the same with
in the tree level. However, in the loop level, we may need to calculate
and others which are more complicated, these will be different in the case of .