Color ordering and partial amplitudes

Choose fundamental SU(N) generators normalized so their pairwise trace equals the color Kronecker delta. Structure constants and generator products then allow a pure-gluon tree amplitude to be expanded in single traces. Each trace multiplies a color-ordered partial amplitude and the common coupling power. The generator normalization is a convention: changing it requires a compensating change of couplings and partial-amplitude normalization, rather than a new physical prediction.

Cyclic rotations leave a trace unchanged, so the trace sum uses permutations modulo cyclic rotations. Reflection and further amplitude identities can reduce the number of independent kinematic functions beyond this first organizational count. The trace expansion is useful even when it is redundant. At loop level multiple-trace structures and distinctions between primitive and partial amplitudes enter; the simple tree single-trace formula must not be presented as the complete loop color decomposition.

An internal line of a planar ordered tree separates consecutive external labels from their cyclic complement. Its momentum is the sum of that consecutive set, so only such sums can generate factorization poles in the partial amplitude. At four points ordered as one, two, three, four, the two distinct channels are the adjacent-pair invariants twelve and twenty-three. The nonadjacent thirteen channel belongs to other orderings rather than this partial amplitude.

At higher multiplicity a set and its complementary consecutive set describe the same internal line because total incoming momentum vanishes. Counting both as separate poles would double count a channel. An allowed channel can still have zero residue for a particular helicity assignment or theory. Conversely, an apparent forbidden pole in an intermediate expression must cancel from the final ordered function; its presence in one algebraic term is not yet proof of a physical singularity.

Each partial amplitude is gauge invariant on shell when the color decomposition is performed consistently. This makes it a useful computational target even though it is not a directly measured color-resolved probability. To form a squared full amplitude, reconstruct the color tensor and contract it with its complex conjugate. Cross terms between different trace structures generally survive. Dropping them requires a stated approximation, such as a controlled leading-color treatment in a suitable setting.

Coupling powers, helicity sums, incoming-state averages, phase-space measures, and infrared-safe measurement functions are additional ingredients of an experimental prediction. None is supplied by merely counting traces. Color ordering instead gives a disciplined analytic workspace: neighboring soft factors, collinear limits, and recursion channels all refer to the same cyclic list. Reordering labels to simplify one expression without transforming its color coefficient can therefore change the reconstructed result.