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BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory

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

TL;DR

The paper provides a field-theoryproof of Bern–Carrasco–Johansson's KLT relation by first establishing KK and BCJ relations for color-ordered scalar φ^3 amplitudes via BCFW recursion with nonzero boundary contributions and by deriving an off-shell fundamental BCJ relation. It then uses these results to prove that tree-level gluon amplitudes can be written as a product of a color-ordered gauge amplitude and a color-ordered scalar amplitude, with two complementary proofs: a direct BCFW-based argument and an off-shell BCJ-based derivation. The work clarifies the role of Jacobi identities in BCJ relations, extends KK/BCJ to scalar theories, and strengthens the understanding of color-kinematics duality, offering tools potentially extensible to loop-level and mixed-p particle configurations.

Abstract

We present a field theoretical proof of the conjectured KLT relation which states that the full tree-level scattering amplitude of gluons can be written as a product of color-ordered amplitude of gluons and color-ordered amplitude of scalars with only cubic vertex. To give a proof we establish the KK relation and BCJ relation of color-ordered scalar amplitude using BCFW recursion relation with nonzero boundary contributions. As a byproduct, an off-shell version of fundamental BCJ relation is proved, which plays an important role in our work.

BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory

TL;DR

The paper provides a field-theoryproof of Bern–Carrasco–Johansson's KLT relation by first establishing KK and BCJ relations for color-ordered scalar φ^3 amplitudes via BCFW recursion with nonzero boundary contributions and by deriving an off-shell fundamental BCJ relation. It then uses these results to prove that tree-level gluon amplitudes can be written as a product of a color-ordered gauge amplitude and a color-ordered scalar amplitude, with two complementary proofs: a direct BCFW-based argument and an off-shell BCJ-based derivation. The work clarifies the role of Jacobi identities in BCJ relations, extends KK/BCJ to scalar theories, and strengthens the understanding of color-kinematics duality, offering tools potentially extensible to loop-level and mixed-p particle configurations.

Abstract

We present a field theoretical proof of the conjectured KLT relation which states that the full tree-level scattering amplitude of gluons can be written as a product of color-ordered amplitude of gluons and color-ordered amplitude of scalars with only cubic vertex. To give a proof we establish the KK relation and BCJ relation of color-ordered scalar amplitude using BCFW recursion relation with nonzero boundary contributions. As a byproduct, an off-shell version of fundamental BCJ relation is proved, which plays an important role in our work.

Paper Structure

This paper contains 14 sections, 68 equations, 9 figures.

Table of Contents

  1. Introduction
  2. BCFW recursion relation and Boundary contribution in $\phi^{3}$ theory
  3. Color-order reversed relation and $U(1)$-decoupling relation
  4. The KK relation for color-ordered scalar theory
  5. Fundamental BCJ relations of color-dressed scalar amplitudes
  6. The pole part
  7. the generalized fundamental BCJ relation with one off-shell momentum
  8. Boundary terms
  9. Field theory proof of color KLT relations
  10. The KLT relation
  11. The proof of KLT relation for gauge theory
  12. Another proof by off-shell BCJ relations
  13. Conclusion
  14. The generalized $U(1)$-decoupling identity

Figures (9)

  • Figure 1: The graph representation of off-shell BCJ relation.
  • Figure 2: The boundary contributions of four-point BCJ relation.
  • Figure 3: The combination of (a) and (f) of Figure 2.
  • Figure 4: The off-shell BCJ relation of four-point amplitude.
  • Figure 5: The Feynman diagrams organized according to the vertex leg $n$ attached.
  • ...and 4 more figures