Gauge and Gravity Amplitude Relations

TL;DR

<p>Gauge and Gravity Amplitude Relations</p> develops a graph-based framework to unify gauge and gravity scattering amplitudes through color-kinematics duality and the double-copy construction. By organizing loop integrands around cubic graphs and enforcing Jacobi-like relations on kinematic numerators, the work demonstrates how gravity amplitudes can be obtained as products of gauge-theory numerators (the double-copy) and how KLT relations emerge as a momentum-kernel reformulation at tree level. The methodology hinges on unitarity-based verification and the maximal-cut construction to fix numerators while preserving locality and power counting, with special emphasis on ${\mathcal{N}}=4$ super Yang–Mills and related theories. The results illuminate dramatic reductions in the space of independent building blocks (master numerators) and establish a scalable path toward all-multiplicity, multi-loop calculations, with implications for both perturbative gravity and high-precision collider predictions.</p>

Abstract

In these lectures I talk about simplifications and universalities found in scattering amplitudes for gauge and gravity theories. In contrast to Ward identities, which are understood to arise from familiar symmetries of the classical action, these structures are currently only understood in terms of graphical organizational principles, such as the gauge-theoretic color-kinematics duality and the gravitational double-copy structure, for local representations of multi-loop S-matrix elements. These graphical principles make manifest new relationships in and between gauge and gravity scattering amplitudes. My lectures will focus on arriving at such graphical organizations for generic theories with examples presented from maximal supersymmetry, and their use in unitarity-based multi-loop integrand construction.

Figures (6)

• Figure 1: Comparison of various scaling behaviors. Quadratic scaling, $m^2$, is associated with the number of interaction terms for classical $m$-body systems with two-body interactions. Note that the far sharper exponential scaling, $e^m$, associated with the number of cubic graphs contributing to a color-ordered tree-level amplitude, is still subdominant for $m>7$ to the factorial scaling associated with the number of cubic graphs associated with $m$-particle tree-level interaction, $(2m-5)!!$, which itself is subdominant to the number of terms in the smallest known Kawai-Lewellen-Tye (KLT) representation of generic tree-level gravity amplitudes $((m-3)!)^2$.
• Figure 2: Half-ladder tree-level graph.
• Figure 3: The tree graphs contributing from each of the trees in the planar 2-loop cut eq. (\ref{['cutpDefn']}).
• Figure 4: The hierarchy of cuts considered in the method of maximal cuts for a two-loop four-point amplitude. Every exposed internal leg is taken to be cut, every blob is meant to represent a tree. First one considers the maximal cuts, then the next-to-maximal cuts and so on. Note that the next${}^n$-to-maximal cuts involve higher-point trees.
• Figure 5: When a multi-loop graph has an automorphism, the numerator must satisfy automorphism symmetry at least up to the redundancy of the theory. For ${\@fontswitch\mathcal{N}}=4$ super-Yang-Mills the above automorphism holds for all external states -- i.e. the same function applies to every labeling of the same topology that contributes to the integrand. For pure Yang-Mills the above relation only holds when all four external gluons have the same helicity.

Theorems & Definitions (1)

• Example 1