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Dual-color decompositions at one-loop level in Yang-Mills theory

Yi-Jian Du, Bo Feng, Chih-Hao Fu

TL;DR

This work extends color-kinematic duality to one-loop Yang-Mills amplitudes by formulating two dual color decompositions: dual DDM-form and dual trace-form. The dual DDM-form expresses the integrand as a sum over color-ordered scalar-loop building blocks with dual kinematic numerators, calculable via Jacobi-like identities from the BCJ framework, with explicit 2-, 3-, and 4-point examples illustrating the construction. The dual-trace-form maps the kinematic numerators to cyclic, reflection-symmetric kinematic traces au using KK-relations and their relation to the dual DDM structure, supported by detailed positive-definite G-matrix constructions up to six points and an alternative symmetry-based method. Collectively, the paper provides a coherent, algorithmic approach to one-loop dual-color decompositions and demonstrates how the dual forms reproduce known results while offering practical pathways for higher-point computations.

Abstract

In this work, we extend the construction of dual color decomposition in Yang-Mills theory to one-loop level, i.e., we show how to write one-loop integrands in Yang-Mills theory to the dual DDM-form and the dual trace-form. In dual forms, integrands are decomposed in terms of color-ordered one-loop integrands for color scalar theory with proper dual color coefficients.In dual DDM decomposition, The dual color coefficients can be obtained directly from BCJ-form by applying Jacobi-like identities for kinematic factors. In dual trace decomposition, the dual trace factors can be obtained by imposing one-loop KK relations, reflection relation and their relation with the kinematic factors in dual DDM-form.

Dual-color decompositions at one-loop level in Yang-Mills theory

TL;DR

This work extends color-kinematic duality to one-loop Yang-Mills amplitudes by formulating two dual color decompositions: dual DDM-form and dual trace-form. The dual DDM-form expresses the integrand as a sum over color-ordered scalar-loop building blocks with dual kinematic numerators, calculable via Jacobi-like identities from the BCJ framework, with explicit 2-, 3-, and 4-point examples illustrating the construction. The dual-trace-form maps the kinematic numerators to cyclic, reflection-symmetric kinematic traces au using KK-relations and their relation to the dual DDM structure, supported by detailed positive-definite G-matrix constructions up to six points and an alternative symmetry-based method. Collectively, the paper provides a coherent, algorithmic approach to one-loop dual-color decompositions and demonstrates how the dual forms reproduce known results while offering practical pathways for higher-point computations.

Abstract

In this work, we extend the construction of dual color decomposition in Yang-Mills theory to one-loop level, i.e., we show how to write one-loop integrands in Yang-Mills theory to the dual DDM-form and the dual trace-form. In dual forms, integrands are decomposed in terms of color-ordered one-loop integrands for color scalar theory with proper dual color coefficients.In dual DDM decomposition, The dual color coefficients can be obtained directly from BCJ-form by applying Jacobi-like identities for kinematic factors. In dual trace decomposition, the dual trace factors can be obtained by imposing one-loop KK relations, reflection relation and their relation with the kinematic factors in dual DDM-form.

Paper Structure

This paper contains 12 sections, 80 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Dual DDM-form
  3. Two-point example
  4. Three-point example
  5. Four-point example
  6. Dual trace-form
  7. general structure of the defining conditions
  8. Four-point dual traces
  9. Five-point case
  10. Six-point example
  11. An alternative approach
  12. Conclusion

Figures (5)

  • Figure 1: Feynman-like diagrams for two-point one-loop integrand using only cubic vertex.
  • Figure 2: Feynman-like diagrams for three-point one-loop integrand.
  • Figure 3: Feynman-like diagrams with two lines connected to the loop in four-point case.
  • Figure 4: Feynman-like diagrams with three lines connected to the loop in four-point case.
  • Figure 5: Feynman-like diagrams with four lines connected to the loop in four-point case.