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Color structures and permutations

Barak Kol, Ruth Shir

TL;DR

This work develops a rigorous, representation-theoretic framework for color structures in tree-level gauge-theory amplitudes. It identifies the space of color structures with the cyclic Lie operad and derives the permutation-character via a generating function, revealing the irreducible content for small n and reducibility for larger n. It also bridges the common f-based and trace-based color decompositions, proving Kleiss–Kuijf relations for color-ordered sub-amplitudes and showing that Parke–Taylor and CHY factors obey these relations through Grassmannian structure. The results provide a principled basis for analyzing sub-amplitude symmetries and guide efficient construction of amplitudes across gauge theories.

Abstract

Color structures for tree level scattering amplitudes in gauge theory are studied in order to determine the symmetry properties of the color-ordered sub-amplitudes. We mathematically formulate the space of color structures together with the action of permuting external legs. The character generating functions are presented from the mathematical literature and we determine the decomposition into irreducible representations. Mathematically, free Lie algebras and the Lie operad are central. A study of the implications for sub-amplitudes is initiated and we prove directly that both the Parke-Taylor amplitudes and Cachazo-He-Yuan amplitudes satisfy the Kleiss-Kuijf relations.

Color structures and permutations

TL;DR

This work develops a rigorous, representation-theoretic framework for color structures in tree-level gauge-theory amplitudes. It identifies the space of color structures with the cyclic Lie operad and derives the permutation-character via a generating function, revealing the irreducible content for small n and reducibility for larger n. It also bridges the common f-based and trace-based color decompositions, proving Kleiss–Kuijf relations for color-ordered sub-amplitudes and showing that Parke–Taylor and CHY factors obey these relations through Grassmannian structure. The results provide a principled basis for analyzing sub-amplitude symmetries and guide efficient construction of amplitudes across gauge theories.

Abstract

Color structures for tree level scattering amplitudes in gauge theory are studied in order to determine the symmetry properties of the color-ordered sub-amplitudes. We mathematically formulate the space of color structures together with the action of permuting external legs. The character generating functions are presented from the mathematical literature and we determine the decomposition into irreducible representations. Mathematically, free Lie algebras and the Lie operad are central. A study of the implications for sub-amplitudes is initiated and we prove directly that both the Parke-Taylor amplitudes and Cachazo-He-Yuan amplitudes satisfy the Kleiss-Kuijf relations.

Paper Structure

This paper contains 13 sections, 51 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Color trees and traces
  3. f-based color trees
  4. Traces
  5. Relating the f and trace decompositions
  6. Section conclusions
  7. Action by permutations
  8. Free Lie algebras and the Lie operad
  9. The nature of $TCS_n$
  10. Implications for sub-amplitudes
  11. Conclusions
  12. Definition of Operads
  13. Irreducible $S_n$ components

Figures (1)

  • Figure 1: An example of Jacobi planting, whereby any color diagram can be converted to a combination of flat or multi-peripheral diagrams through a repeated use of the Jacobi identity (\ref{['Jacobi']}).