Gravity as the Square of Gauge Theory

TL;DR

This paper investigates the duality between color and kinematics in gauge theories, demonstrating how gravity amplitudes can be derived from gauge-theory numerators through a squaring relation, extending to loop levels.

Contribution

It provides a field-theory proof of the duality, constructs a Yang-Mills Lagrangian manifesting the duality, and shows how gravity amplitudes can be obtained by squaring gauge-theory numerators.

Findings

Gravity amplitudes are the product of two gauge-theory numerators.

A Yang-Mills Lagrangian manifesting the duality is constructed.

The duality extends to loop amplitudes and higher points.

Abstract

We explore consequences of the recently discovered duality between color and kinematics, which states that kinematic numerators in a diagrammatic expansion of gauge-theory amplitudes can be arranged to satisfy Jacobi-like identities in one-to-one correspondence to the associated color factors. Using on-shell recursion relations, we give a field-theory proof showing that the duality implies that diagrammatic numerators in gravity are just the product of two corresponding gauge-theory numerators, as previously conjectured. These squaring relations express gravity amplitudes in terms of gauge-theory ingredients, and are a recasting of the Kawai, Lewellen and Tye relations. Assuming that numerators of loop amplitudes can be arranged to satisfy the duality, our tree-level proof immediately carries over to loop level via the unitarity method. We then present a Yang-Mills Lagrangian whose diagrams through five points manifestly satisfy the duality between color and kinematics. The existence of such Lagrangians suggests that the duality also extends to loop amplitudes, as confirmed at two and three loops in a concurrent paper. By "squaring" the novel Yang-Mills Lagrangian we immediately obtain its gravity counterpart. We outline the general structure of these Lagrangians for higher points. We also write down various new representations of gauge-theory and gravity amplitudes that follow from the duality between color and kinematics.