The Duality Between Color and Kinematics and its Applications

TL;DR

The review articulates color-kinematics (CK) duality and the BCJ double-copy framework as a unifying method to recast gravity amplitudes in terms of gauge-theory building blocks. It explains how CK-dual numerators enable gravity amplitudes via color-to-kinematics replacement and the KLT relations, extends these ideas to loop level, and maps a broad web of double-copy-constructible theories including YM, YME, gauged/conformal supergravities, and perturbative string theories. The work emphasizes geometric and boundary-data organization of amplitudes, inheritance of symmetries, and the classical double copy, highlighting practical rules and constructive approaches (orbifolds, Higgsing, masses) for generating new theories. It also surveys explicit loop-level examples and the role of soft limits in testing enhanced symmetries, underscoring the significance of CK duality for simplifying gravity computations and for revealing deep connections between gauge theories and gravity across diverse physical settings.

Abstract

This review describes the duality between color and kinematics and its applications, with the aim of gaining a deeper understanding of the perturbative structure of gauge and gravity theories. We emphasize, in particular, applications to loop-level calculations, the broad web of theories linked by the duality and the associated double-copy structure, and the issue of extending the duality and double copy beyond scattering amplitudes. The review is aimed at doctoral students and junior researchers both inside and outside the field of amplitudes and is accompanied by various exercises.

Figures (37)

• Figure 1: Gauge theories have three- and four-point vertices in a Feynman diagrammatic description.
• Figure 2: Gravity theories have an infinite number of higher-point contact interactions in a Feynman diagrammatic description.
• Figure 3: The three Feynman diagrams corresponding to the $s$, $t$ and $u$ channels.
• Figure 4: Connections of CK duality to various topics. This review will discuss in some detail the connection of CK duality to the topics in the upper right (with the main chapters indicated) and less so to the topics on the lower right. The various topics are intertwined with each other as well.
• Figure 5: Color-algebra relations in the adjoint (a) and fundamental representation (b). The curly lines represent adjoint representation states and the straight lines fundamental representation. The vertices correspond to the color matrices in Eq. (\ref{['ftildeNorm']}).