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Unifying Relations for Scattering Amplitudes

Clifford Cheung, Chia-Hsien Shen, Congkao Wen

TL;DR

This work introduces a set of Lorentz-invariant differential operators that transmute tree-level scattering amplitudes into amplitudes of a broad web of theories, uncovering a unified structure linking YM, gravity, BI, DBI, SG, NLSM, and BS in arbitrary dimensions. By starting from extended gravity (gravity with a dilaton and two-form) and applying trace, insertion, and longitudinal operators, the authors generate the S-matrices of diverse theories and show that core structures like KLT, BCJ, and CHY naturally descend through transmutation. They provide a constructive, inductive proof that these relations hold at all multiplicities, analyze factorization and large-$z$ behavior, and establish a coherent infrared picture where soft theorems and Adler zeros are inherited across the web. The framework also accommodates UV completions and connects to string-theory amplitudes via open-string and Z-theory constructions, suggesting deep, practical ways to transport known results across theories. Overall, transmutation offers a powerful, dimension-agnostic lens on the unity of scattering amplitudes and their symmetries, with potential extensions to fermions and loops.

Abstract

We derive new amplitudes relations revealing a hidden unity among wide-ranging theories in arbitrary spacetime dimensions. Our results rely on a set of Lorentz invariant differential operators which transmute physical tree-level scattering amplitudes into new ones. By transmuting the amplitudes of gravity coupled to a dilaton and two-form, we generate all the amplitudes of Einstein-Yang-Mills theory, Dirac-Born-Infield theory, special Galileon, nonlinear sigma model, and biadjoint scalar theory. Transmutation also relates amplitudes in string theory and its variants. As a corollary, celebrated aspects of gluon and graviton scattering like color-kinematics duality, the KLT relations, and the CHY construction are inherited traits of the transmuted amplitudes. Transmutation recasts the Adler zero as a trivial consequence of the Weinberg soft theorem and implies new subleading soft theorems for certain scalar theories.

Unifying Relations for Scattering Amplitudes

TL;DR

This work introduces a set of Lorentz-invariant differential operators that transmute tree-level scattering amplitudes into amplitudes of a broad web of theories, uncovering a unified structure linking YM, gravity, BI, DBI, SG, NLSM, and BS in arbitrary dimensions. By starting from extended gravity (gravity with a dilaton and two-form) and applying trace, insertion, and longitudinal operators, the authors generate the S-matrices of diverse theories and show that core structures like KLT, BCJ, and CHY naturally descend through transmutation. They provide a constructive, inductive proof that these relations hold at all multiplicities, analyze factorization and large- behavior, and establish a coherent infrared picture where soft theorems and Adler zeros are inherited across the web. The framework also accommodates UV completions and connects to string-theory amplitudes via open-string and Z-theory constructions, suggesting deep, practical ways to transport known results across theories. Overall, transmutation offers a powerful, dimension-agnostic lens on the unity of scattering amplitudes and their symmetries, with potential extensions to fermions and loops.

Abstract

We derive new amplitudes relations revealing a hidden unity among wide-ranging theories in arbitrary spacetime dimensions. Our results rely on a set of Lorentz invariant differential operators which transmute physical tree-level scattering amplitudes into new ones. By transmuting the amplitudes of gravity coupled to a dilaton and two-form, we generate all the amplitudes of Einstein-Yang-Mills theory, Dirac-Born-Infield theory, special Galileon, nonlinear sigma model, and biadjoint scalar theory. Transmutation also relates amplitudes in string theory and its variants. As a corollary, celebrated aspects of gluon and graviton scattering like color-kinematics duality, the KLT relations, and the CHY construction are inherited traits of the transmuted amplitudes. Transmutation recasts the Adler zero as a trivial consequence of the Weinberg soft theorem and implies new subleading soft theorems for certain scalar theories.

Paper Structure

This paper contains 21 sections, 104 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. Transmutation
  3. Physical Constraints
  4. Transmutation Operators
  5. Unified Web of Theories
  6. Single Trace Amplitudes
  7. Multiple Trace Amplitudes
  8. Summary of Unifying Relations
  9. Ultraviolet Completion
  10. Examples
  11. Three-Particle Amplitudes
  12. Four-Particle Amplitudes
  13. Gluon and Graviton Amplitudes
  14. Proof by Induction
  15. On-shell Constructibility
  16. ...and 6 more sections

Figures (1)

  • Figure 1: Diagram depicting the unified web of theories. The corners represent extended gravity (G), Einstein-Maxwell (EM) theory, Yang-Mills (YM) theory, Born-Infeld (BI) theory, Dirac-Born-Infeld (DBI) scalar theory, nonlinear sigma model (NLSM), special Galileon (SG), Yang-Mills scalar (YMS) theory, and biadjoint (BS) theory. The arrows correspond to the transmutation operators, ${\cal T}_{ij}$, ${\cal T}_{ijk}$, and ${\cal L}$. The shaded regions and edges correspond to hybrid theories.