New Color Decompositions for Gauge Amplitudes at Tree and Loop Level

TL;DR

The paper tackles the color decomposition problem in QCD amplitudes by introducing a structure-constant-based decomposition for gluon tree amplitudes and one-loop amplitudes, reducing to a minimal set of independent subamplitudes. Two independent proofs show equivalence to the Kleiss-Kuijff relation and establish the correspondence of kinematic coefficients with color-ordered subamplitudes. Extensions to one-loop amplitudes with external quark-antiquark pairs are provided, including leading and subleading color contributions and NLO/NNLO interference formulas. The approach yields computational advantages for cross sections and jet-rate predictions, and is compatible with existing trace-based formalisms while exposing clearer color structure.

Abstract

Recently, a color decomposition using structure constants was introduced for purely gluonic tree amplitudes, in a compact form involving only the linearly independent subamplitudes. We give two proofs that this decomposition holds for an arbitrary number of gluons. We also present and prove similar decompositions at one loop, both for pure gluon amplitudes and for amplitudes with an external quark-antiquark pair.

Figures (6)

• Figure 1: Graphical representation of a multi-peripheral color factor. A vertex stands for $f^{abc}$, and a bond for $\delta^{ab}$.
• Figure 2: The Jacobi identity for structure constants in graphical notation.
• Figure 3: Graphical representation of the color factor for a generic tree-level Feynman diagram. Also shown is a step in its conversion to multi-peripheral form, by using the Jacobi identity in the dashed region.
• Figure 4: Contraction of different multi-peripheral color factors with a single $\lambda$ trace. The heavy line with an arrow denotes the fundamental representation. The left diagram shows the contraction where $a_i$, $i=2,3,\ldots,n-1$, appear in the same order in the $\lambda$ trace as in the multi-peripheral color factor. Every other contraction gives rise to a nonplanar (and hence color-suppressed) diagram of the type shown on the right.
• Figure 5: (a) Ring color factors for one-loop $n$-gluon amplitudes. (b) The corresponding color factors for one-loop amplitudes with an external quark-antiquark pair.