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On Tree Amplitudes in Gauge Theory and Gravity

Nima Arkani-Hamed, Jared Kaplan

TL;DR

This work provides a dimension-agnostic, physical understanding of why BCFW recursion works for tree amplitudes in gauge theory and gravity. By interpreting large-z deformations as a hard light-like probe in a soft background and exploiting the background-field method in light-cone gauge, Arkani-Hamed and Kaplan reveal an enhanced spin Lorentz symmetry that constrains leading large-z behavior. They show that gluon and graviton amplitudes can vanish at infinity for many helicities, enabling BCFW recursions in any number of dimensions and clarifying the structure of scalar, photon, and graviton interactions. The results have implications for higher-dimensional techniques, KK reductions, and possible loop-level cancellations, tying together S-matrix reasoning with explicit Lagrangian analysis.

Abstract

The BCFW recursion relations provide a powerful way to compute tree amplitudes in gauge theories and gravity, but only hold if some amplitudes vanish when two of the momenta are taken to infinity in a particular complex direction. This is a very surprising property, since individual Feynman diagrams all diverge at infinite momentum. In this paper we give a simple physical understanding of amplitudes in this limit, which corresponds to a hard particle with (complex) light-like momentum moving in a soft background, and can be conveniently studied using the background field method exploiting background light-cone gauge. An important role is played by enhanced spin symmetries at infinite momentum--a single copy of a "Lorentz" group for gauge theory and two copies for gravity--which together with Ward identities give a systematic expansion for amplitudes at large momentum. We use this to study tree amplitudes in a wide variety of theories, and in particular demonstrate that certain pure gauge and gravity amplitudes do vanish at infinity. Thus the BCFW recursion relations can be used to compute completely general gluon and graviton tree amplitudes in any number of dimensions. We briefly comment on the implications of these results for computing massive 4D amplitudes by KK reduction, as well understanding the unexpected cancelations that have recently been found in loop-level gravity amplitudes.

On Tree Amplitudes in Gauge Theory and Gravity

TL;DR

This work provides a dimension-agnostic, physical understanding of why BCFW recursion works for tree amplitudes in gauge theory and gravity. By interpreting large-z deformations as a hard light-like probe in a soft background and exploiting the background-field method in light-cone gauge, Arkani-Hamed and Kaplan reveal an enhanced spin Lorentz symmetry that constrains leading large-z behavior. They show that gluon and graviton amplitudes can vanish at infinity for many helicities, enabling BCFW recursions in any number of dimensions and clarifying the structure of scalar, photon, and graviton interactions. The results have implications for higher-dimensional techniques, KK reductions, and possible loop-level cancellations, tying together S-matrix reasoning with explicit Lagrangian analysis.

Abstract

The BCFW recursion relations provide a powerful way to compute tree amplitudes in gauge theories and gravity, but only hold if some amplitudes vanish when two of the momenta are taken to infinity in a particular complex direction. This is a very surprising property, since individual Feynman diagrams all diverge at infinite momentum. In this paper we give a simple physical understanding of amplitudes in this limit, which corresponds to a hard particle with (complex) light-like momentum moving in a soft background, and can be conveniently studied using the background field method exploiting background light-cone gauge. An important role is played by enhanced spin symmetries at infinite momentum--a single copy of a "Lorentz" group for gauge theory and two copies for gravity--which together with Ward identities give a systematic expansion for amplitudes at large momentum. We use this to study tree amplitudes in a wide variety of theories, and in particular demonstrate that certain pure gauge and gravity amplitudes do vanish at infinity. Thus the BCFW recursion relations can be used to compute completely general gluon and graviton tree amplitudes in any number of dimensions. We briefly comment on the implications of these results for computing massive 4D amplitudes by KK reduction, as well understanding the unexpected cancelations that have recently been found in loop-level gravity amplitudes.

Paper Structure

This paper contains 12 sections, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. BCFW Redux
  3. Surprising Behavior of $M(z)$ as $z \to \infty$
  4. Understanding $M(z \to \infty)$ in YM and GR
  5. Scalar QED Amplitudes
  6. Dominant Large $z$ Behavior
  7. Scalar-Yang-Mills Amplitudes
  8. Gluon Amplitudes and Enhanced Spin Symmetry as $z \to \infty$
  9. Scalar-Graviton Amplitudes
  10. Photon-Graviton Amplitudes
  11. Amplitudes in General Relativity
  12. Discussion

Figures (3)

  • Figure 1: The BCFW recursion relation computes an $n$-point amplitude by sewing together lower-point amplitudes with (complex) on-shell momenta.
  • Figure 2: Contributions to the analytically continued amplitudes $M(z)$ from individual Feynman diagrams diverge as $z \to \infty$ for gauge theories and gravity. This is due to the vertices that grow as $z,z^2$ for gauge theory and gravity, which overcompensate for the $1/z$ scaling of propagators.
  • Figure 3: The unique set of diagrams, for which light-cone gauge is singular, and which dominate large $z$ amplitudes.