On Tree Amplitudes in Gauge Theory and Gravity (0801.2385v1)

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.

• The paper demonstrates that BCFW recursion and enhanced spin symmetries provide a unified method to compute tree-level amplitudes in gauge and gravity theories.
• It shows that for specific helicity configurations, amplitudes vanish at infinite momentum, revealing nontrivial cancellations among divergent Feynman diagrams.
• The findings have implications for higher-dimensional models, enabling derivation of massive 4D amplitudes via Kaluza-Klein reduction from massless theories.

Overview of "On Tree Amplitudes in Gauge Theory and Gravity"

The paper "On Tree Amplitudes in Gauge Theory and Gravity" by Nima Arkani-Hamed and Jared Kaplan addresses the computational techniques for determining tree-level amplitudes in gauge theories and gravity, focusing specifically on the BCFW recursion relations. These recursion relations are a powerful tool for calculating tree amplitudes, assuming that certain amplitudes vanish as two momenta become infinite in a specific complex direction. This vanishing is a nontrivial property since individual Feynman diagrams can diverge at infinite momentum.

Key Concepts and Methodology

The core objective of the paper is to provide a physical understanding of tree amplitudes in the limit where momenta go to infinity. This scenario is studied using a specific gauge choice in the background field method, exploiting background light-cone gauge. The enhanced spin symmetries at infinite momentum, analogous to a Lorentz group for gauge theories and double that for gravity, are crucial. These symmetries, alongside Ward identities, facilitate a systematic expansion of amplitudes at large momentum.

Main Contributions

1. BCFW Recursion and Symmetries: The paper demonstrates that the BCFW recursion relations can be applied to compute tree amplitudes in gauge theories and gravity in any number of dimensions. The authors utilize an enhanced understanding of amplitudes' behavior at large momentum, exploiting symmetries and gauge choices to elucidate this behavior.
2. Behavior of Amplitudes at Infinity: Through explicit examples, the authors show that for specific helicity configurations, amplitudes vanish as momenta become large. This is a significant result as it not only validates the applicability of the BCFW recursion relations but also highlights the intricate cancellations that occur in the expansion of Feynman diagrams.
3. Implications for Theories with Extra Dimensions: The findings have particular relevance for scenarios involving higher-dimensional theories, such as those involving Kaluza-Klein (KK) reduction, where massive 4D amplitudes can be derived from higher-dimensional massless ones.

Theoretical Implications

The implications of these results extend beyond the technical improvement of computation methods for tree amplitudes. The understanding of amplitude behavior at infinite momentum provides insights into the simplifications embedded within gauge and gravity theories. The intriguing symmetry properties discovered suggest deeper underlying principles at play in field theory.

Future Directions and Speculations

The paper's analysis suggests potential avenues for further exploration, particularly concerning the consistency and computation of loop-level amplitudes, as well as the unexpected cancellations observed in gravitational theories at loop level. These findings might hint at more profound symmetries that could influence our understanding of quantum field theories and gravitational interactions.

In conclusion, the paper significantly enhances our understanding of tree-level amplitudes by leveraging sophisticated techniques in gauge symmetry and complex analysis. This work paves the way for refined computational approaches in particle physics and continues to spur theoretical investigations into the structure of fundamental interactions.