Color-flow decomposition of QCD amplitudes

TL;DR

The paper introduces a color-flow decomposition for tree-level multi-parton QCD amplitudes by treating gluons as $N\times N$ matrices, yielding a physically intuitive color-flow structure and color-flow Feynman rules free of fundamental-representation matrices and structure constants. It demonstrates that all-gluon and quark-containing amplitudes decompose into a sum of color flows with simple $(-1/N)^k$ coefficients, and that partial amplitudes can be computed efficiently using planar color-flow diagrams. The approach significantly speeds up multi-jet calculations (e.g., for $n=12$ gluons) and adapts naturally to merging with shower Monte Carlo programs, with subleading contributions handled via straightforward extensions. The method promises faster event generation for collider backgrounds and is slated for integration into MadEvent/MadGraph, extending to processes with leptons, photons, and electroweak bosons.

Abstract

We introduce a new color decomposition for multi-parton amplitudes in QCD, free of fundamental-representation matrices and structure constants. This decomposition has a physical interpretation in terms of the flow of color, which makes it ideal for merging with shower Monte-Carlo programs. The color-flow decomposition allows for very efficient evaluation of amplitudes with many quarks and gluons, many times faster than the standard color decomposition based on fundamental-representation matrices. This will increase the speed of event generators for multi-jet processes, which are the principal backgrounds to signals of new physics at colliders.

Figures (11)

• Figure 1: Color-flow Feynman rules. All momenta are outgoing. The arrows indicate the flow of color. The sum in the three-gluon vertex is over the two non-cyclic permutations of (1,2,3); in the four-gluon vertex, the sum is over the six non-cyclic permutations of (1,2,3,4). When calculating a partial amplitude the sum is dropped, as only one term in the sum contributes to a given color flow.
• Figure 2: The $SU(N)$ gluon propagator may be split into a $U(N)$ gluon propagator and a $U(1)$ gluon propagator. The $U(1)$ gluon interacts only with quarks.
• Figure 3: Color flow $\delta^{i_1}_{j_2} \delta^{i_2}_{j_3} \cdots \delta^{i_n}_{j_1}$. Each pair of indices $i_k,j_k$ corresponds to an external gluon.
• Figure 4: Feynman diagrams corresponding to a four-gluon partial amplitude.
• Figure 5: The color flow $\delta^{i_q}_{j_1} \delta^{i_1}_{j_2} \cdots \delta^{i_n}_{j_q}$ for one $\bar{q}q$ pair and $n$ gluons.