Normalize color before counting radiation

Choose Hermitian traceless fundamental generators with trace of a generator product equal to one half times the adjoint Kronecker delta. Their commutator defines real structure constants. The sum of squared generators commutes with every group transformation and is proportional to the identity in an irreducible representation. In the fundamental representation its coefficient is called C F. Taking the trace of that identity counts all N squared minus one generators, immediately determining C F without selecting a particular color state.

The adjoint representation acts on the gluon index itself. Contracting two structure constants gives the adjoint Casimir C A, which equals N in this normalization. For QCD there are three fundamental components and eight adjoint components, giving C F equal to four thirds and C A equal to three. These values are group identities. They do not count four thirds of a particle or say that a gluon carries a larger continuously variable electric charge.

A useful completeness identity replaces a contracted pair of generators by two color-line contractions. The first exchanges fundamental indices. The second subtracts the trace component, because SU(N) excludes the extra identity generator of U(N). Dropping that second term is a large-N approximation, not an exact manipulation at three colors. This distinction matters when a small interference term is the quantity being studied, even if it is subleading in an inclusive color sum.

A quark and antiquark combine into a singlet plus an octet. The normalized singlet has equal amplitudes for matching color and anticolor, summed coherently. It is not the same density matrix as a random mixture of three matching labels. The amplitude phases are essential for its invariance under a basis rotation. Similarly a three-quark baryon’s simplest color wavefunction uses an antisymmetric invariant tensor; its color structure cannot be inferred merely by adding three classical vectors to zero.

For a single unresolved soft-collinear gluon emitted from a hard parton, a squared color charge multiplies a common leading singular radiation kernel. At identical kinematics that suggests more radiation from an adjoint source than from a fundamental source, with ratio C A over C F. Actual quark and gluon jets also differ through splitting functions, available phase space, flavor mixtures, recoil, higher orders, and hadronization. A measured multiplicity ratio therefore need not equal the bare Casimir ratio.

Confinement means isolated colored asymptotic states are not observed; it does not invalidate their use inside a short-distance factorization calculation. Gauge-invariant hadronic observables connect the perturbative color algebra to experiment. Large-N expansions can organize diagram topologies while missing quantitatively important finite-N effects, and cannot by themselves solve physical QCD at N equal to three. The scattering instrument below only changes a schematic coupling weight. It cannot rotate an SU(3) state or measure a Casimir.