The SAGEX Review on Scattering Amplitudes, Chapter 2: An Invitation to Color-Kinematics Duality and the Double Copy

TL;DR

The SAGEX Chapter 2 investigates whether gravity can be formulated as a double copy of gauge theories via color-kinematics duality, outlining how tree- and loop-level relations (KLT/BCJ) connect gauge-theory amplitudes to gravity amplitudes. It surveys the foundational duality, its extension beyond adjoint representations, and the broad web of double-copy-constructible theories, including ungauged and gauged supergravities and stringy constructions. It then demonstrates how these ideas are used to extract classical gravitational dynamics and gravitational-wave observables from amplitudes, using the 1PM and 2PM results and the generalized unitarity method. The work highlights a unifying framework that links perturbative gravity to a wide class of gauge-theory building blocks and underscores ongoing challenges in identifying the underlying kinematic algebras and expanding the theory web for both quantum and classical gravity applications.

Abstract

Advances in scattering amplitudes have exposed previously-hidden color-kinematics and double-copy structures in theories ranging from gauge and gravity theories to effective field theories such as chiral perturbation theory and the Born-Infeld model. These novel structures both simplify higher-order calculations and pose tantalizing questions related to a unified framework underlying relativistic quantum theories. This introductory mini-review article invites further exploration of these topics. After a brief introduction to color-kinematics duality and the double copy as they emerges at tree and loop-level in gauge and gravity theories, we present two distinct examples: 1) an introduction to the web of double-copy-constructible theories, and 2) a discussion on the application of the double copy to calculation relevant to gravitational-wave physics.

Figures (5)

• Figure 1: The three diagrams with cubic vertices describing a four-point tree amplitude.
• Figure 2: An $m$-point half-ladder tree diagram.
• Figure 3: Graphical representation of the color-algebra relations in the adjoint (a) and some arbitrary representation (b). The curly lines represent adjoint representation states and the straight lines the arbitrary representation.
• Figure 4: Web of double-copy-constructible theories. Undirected links with different colors are drawn between theories that have a common gauge-theory factor. For example, blue: pure SYM theory, red: $(DF)^2$ theory, green: NLSM, pink: (S)YM theory with massless matter, violet: spontaneously broken (S)YM theory. Directed links point toward double-copy constructions that are obtained by modifying both gauge-theory factors. Examples include: adding matter representations, assigning VEVs or truncating/projecting out some states.
• Figure 5: Left-hand side: the part of the four-scalar one-loop amplitude that does not contain intersecting matter lines. Right-hand side: an identification of the part of the four-scalar one-loop amplitude that do not contain intersecting matter lines and have at least one matter line in the loop. Factorization of tree- and loop-level amplitude imply that the shaded blobs are tree-level amplitudes.