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Dual Identities inside the Gluon and the Graviton Scattering Amplitudes

S. -H. Henry Tye, Yang Zhang

TL;DR

This paper uses heterotic and open-string formalisms to refine and prove BCJ-like color–kinematic dualities in M-gluon tree amplitudes, showing how left-moving color factors and right-moving kinematic numerators obey dual Jacobi identities and can be combined via KLT to reproduce YM and gravity amplitudes. It develops a systematic contour-integral approach to generate color and kinematic identities for general M, analyzes gauge-choice issues for the kinematic sector, and demonstrates, through explicit 4- and 5-point examples, that gravity amplitudes can be written diagonally as sums of products of kinematic numerators. The work provides a robust bridge from gauge theory to gravity via string-theoretic constructions and suggests the broader applicability of BCJ dualities to loops and fermionic states. Overall, the study reinforces the deep intertwining of gauge and gravity amplitudes and offers practical pathways to compute them without explicit Feynman rules.

Abstract

Recently, Bern, Carrasco and Johansson conjectured dual identities inside the gluon tree scattering amplitudes. In this paper, we use the properties of the heterotic string and open string tree scattering amplitudes to refine and derive these dual identities. These identities can be carried over to loop amplitudes using the unitarity method. Furthermore, given the $M$-gluon (as well as gluon-gluino) tree amplitudes, $M$-graviton (as well as graviton-gravitino) tree scattering amplitudes can be written down immediately, avoiding the derivation of Feynman rules and the evaluation of Feynman diagrams for graviton scattering amplitudes.

Dual Identities inside the Gluon and the Graviton Scattering Amplitudes

TL;DR

This paper uses heterotic and open-string formalisms to refine and prove BCJ-like color–kinematic dualities in M-gluon tree amplitudes, showing how left-moving color factors and right-moving kinematic numerators obey dual Jacobi identities and can be combined via KLT to reproduce YM and gravity amplitudes. It develops a systematic contour-integral approach to generate color and kinematic identities for general M, analyzes gauge-choice issues for the kinematic sector, and demonstrates, through explicit 4- and 5-point examples, that gravity amplitudes can be written diagonally as sums of products of kinematic numerators. The work provides a robust bridge from gauge theory to gravity via string-theoretic constructions and suggests the broader applicability of BCJ dualities to loops and fermionic states. Overall, the study reinforces the deep intertwining of gauge and gravity amplitudes and offers practical pathways to compute them without explicit Feynman rules.

Abstract

Recently, Bern, Carrasco and Johansson conjectured dual identities inside the gluon tree scattering amplitudes. In this paper, we use the properties of the heterotic string and open string tree scattering amplitudes to refine and derive these dual identities. These identities can be carried over to loop amplitudes using the unitarity method. Furthermore, given the -gluon (as well as gluon-gluino) tree amplitudes, -graviton (as well as graviton-gravitino) tree scattering amplitudes can be written down immediately, avoiding the derivation of Feynman rules and the evaluation of Feynman diagrams for graviton scattering amplitudes.

Paper Structure

This paper contains 21 sections, 162 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Yang-Mills, Heterotic and Open String Scattering Tree Amplitudes
  3. The 4-Gluon Tree Amplitude from the Heterotic String Model
  4. Left-moving amplitudes
  5. Right-moving amplitudes
  6. The Yang-Mills amplitude
  7. M-gluon Tree Scattering Amplitudes
  8. Left-moving amplitudes
  9. Contour integral method and the color (Jacobi) identities
  10. Right-moving amplitudes
  11. Yang-Mills amplitudes
  12. The 5-gluon Tree Amplitude Example
  13. Left-moving amplitudes and color identities
  14. Right-moving amplitude and the gauge choices
  15. KLT relation and the $5$-point amplitudes
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Several examples of the poles $P=\Pi_j s_{ln}$ and the color factors. The structure constants are labeled in the counter-clockwise direction. The field theory tree amplitudes $A^{tree}$ are related to the zero-slope limit of the open string amplitudes $A^{open}$, which are given by the disc diagrams in open string theory. The (yellow) disc for each graph is shown to emphasize this feature.
  • Figure 2: General color (Jacobi) identity for the color factors in tree diagrams. The discs A, B, C and D represent the sub-diagrams.
  • Figure 3: The contour integral over $x_{p+1}$.