Structures and Frontiers in Amplitudes
Scattering amplitudes can reveal algebraic and geometric organization hidden by individual diagrams. This chapter introduces several such structures while keeping their theory, dimension, loop order, and analytic assumptions explicit; a successful construction in one setting does not establish a universal representation.
Enter this chapter
Section titled “Enter this chapter”Begin with Color–Kinematics Duality if cubic graphs and factorization are familiar. For analytic constraints rather than representations, enter through the amplitude-bootstrap page.
Topics
Section titled “Topics”- Color–Kinematics Duality — Tests when kinematic numerators can satisfy color-like algebraic identities.
- The Double Copy — Constructs amplitudes by replacing color factors under specified duality conditions.
- Positive Geometry and Canonical Forms — Relates boundary-residue geometry to canonical differential forms.
- Celestial Amplitudes — Changes the external-state basis using energy-scale transforms.
- S-Matrix and Amplitude Bootstrap — Constrains candidate amplitudes through analyticity, crossing, and unitarity.
- Function Spaces, Symbols, and Coaction Patterns — Organizes selected integral sectors through restricted function spaces and iterated structure.
How the ideas fit together
Section titled “How the ideas fit together”Color–kinematics duality places algebraic restrictions on a graph representation; the double copy uses suitable numerators as replacement data. The resulting spectrum depends on both input state spaces, so a gravity-like output need not be pure Einstein gravity. The original tree-level relations are developed in Bern, Carrasco, and Johansson 2008, §§ II–III.
Positive geometries, celestial transforms, and bootstrap constraints organize different objects. A change of external-state basis does not itself prove a holographic dual, and a symbol can omit constants or lower-weight information. For each method, ask what is reconstructed and which ambiguity its data cannot see.
Review the chapter
Section titled “Review the chapter”Why must the state content be checked after double copying two gauge-theory amplitudes?
Solution
The external states combine through a tensor product. Two vector polarizations can contain graviton, scalar, and antisymmetric-tensor sectors rather than only a graviton. A claim about pure gravity therefore requires the appropriate projection, matter cancellation, or other explicit construction.
Where to go next
Section titled “Where to go next”Continue to The Double Copy for a worked structural application, or S-Matrix and Amplitude Bootstrap for a constraint-based route.