Gravity as the Square of Gauge Theory (1004.0693v1)

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.

• The paper shows that gravitational amplitudes are derived as the square of gauge-theory numerators through established color-kinematics duality.
• It implements on-shell recursion and unitarity methods to extend the proof from tree-level to loop-level amplitudes, confirming BCJ relations.
• The study introduces a novel Yang-Mills Lagrangian with explicit duality and auxiliary fields, paving the way for unified analyses of gravity.

Gravity as the Square of Gauge Theory

In the paper "Gravity as the Square of Gauge Theory," Bern, Dennen, Huang, and Kiermaier explore the intriguing duality between color and kinematics in gauge theories, proposing that gravitational amplitudes can be expressed as products of gauge-theory numerators—a proposition often referred to as the "squaring" relation. This paper builds upon the foundational Kawai-Lewellen-Tye (KLT) relations, which establish connections between tree-level amplitudes in string theory and have been seen to hold in field theories under specific conditions.

The central concept investigated in this paper is the duality that allows kinematic numerators in the expansion of gauge-theory amplitudes to satisfy relations reminiscent of the Jacobi identities obeyed naturally by color factors of Yang-Mills theories. This duality suggests that gravitational amplitudes can be construed as "squares" of gauge-theory amplitudes, thereby simplifying our approach to understanding gravity in terms of the more familiar gauge theories.

The paper presents an explicit field-theory proof of this duality through on-shell recursion relations, demonstrating how it ensures that the gravitational numerators are products of gauge-theory numerators. The duality and the resulting squaring relations are shown not to be restricted to tree-level amplitudes but also extend naturally to loop amplitudes using the unitarity method.

A significant outcome of this work is the presentation of a Yang-Mills Lagrangian that explicitly manifests this color-kinematic duality through diagrams up to five points, reinforcing the viability of the duality even in loop amplitudes, as confirmed independently in related works. The proofs conducted make extensive use of generalized gauge invariance, allowing the authors to manipulate amplitudes such that the numerators adhere to the sought-after duality conditions across all channels, demonstrating a substantial rearrangement of terms permissible in these proofs.

The paper also discusses the implications of these squaring relations for various theories, including pure Yang-Mills and more extended supersymmetrical cases like supersymmetric Yang-Mills/"=8" supergravity theories, illustrating that the findings are not confined to conventional theories of gravity but extend to broader theoretical contexts.

By introducing auxiliary fields within the formalism, the authors manage to construct a non-local but mathematically coherent Lagrangian that respects the BCJ duality. This Lagrangian can, by extension, be "squared" to provide a parallel gravitational theory, thus cementing their proposal that gravity can be considered the squared field of gauge-theory constituents.

While challenges remain, particularly in assembling a complete all-order Lagrangian that universally respects kinematic duality, the present work lays critical groundwork and boldly suggests an underlying simplicity in gauge-gravity relations far more profound than previously recognized. It draws a pathway to exploring gravitational physics with tools native to gauge-theory analysis—potentially opening avenues to greater unification in theoretical physics.

This exploration of duality and its consequences provides potent insights into the intersection of gauge theories and gravity, offering a provocative and mathematically rigorous perspective on longstanding questions in theoretical and high-energy physics. If further substantiated, these results may provide crucial steps towards a more unified understanding of gravity and its place in the quantum field theoretical landscape.