Appendix A:

Color Matrix Identities and Invariants

Only a few identities are necessary for the calculations described in the text. In general, for representation R, SU(N) generators can be picked to satisfy

Tr [ T_a^(R) T_b^(R) ] = T(R) _ab,

with T(R) a number characteristic of the representation. Also of special interest is the representation-dependent invariant, C₂(R), defined by

with I the identity matrix.

We encounter only two representations here, the N-dimensional "defining" representation, F, and the (N²-1)-dimensional adjoint representation, A. The generators T_a^(F) are a complete set of NxN traceless hermitian matrices, while the generators T_a^(A) are defined by the SU(N) structure constants C_abc [Eq. (2.5)] as

[T_a^(A) ]_bc = -i C_abc .

For these two representations, the relevant constants are

Another useful identity, special to the defining representation, enables us to work with simple products of the generators,

with I the 3x3 identity, and the d_abc real. Unlike the previous equations, this and the following equation apply only to SU(3). A numerical value that occurs in the three-loop correction to the total e⁺e^- annihilation cross section is