Two-Loop Scattering Amplitudes: Double-Forward Limit and Colour-Kinematics Duality

TL;DR

This paper extends worldsheet-inspired scattering-equation methods to two loops, proposing all-multiplicity two-loop integrands for Yang–Mills and gravity built as a double-forward limit of tree-level trivalent diagrams. A key advance is the dual representation: a cross-ratio (or slash) CHY-like construction that yields linear propagators and preserves a loop-level colour–kinematics duality derived from tree-level structures. The authors address non-supersymmetric theories via the Neveu–Schwarz sector of ambitwistor strings, relate NS-sector results to supersymmetric cases, and provide explicit checks against maximal unitarity cuts. While the forward-limit framework naturally produces CK duality, translating these representations to conventional Feynman propagators and integrating them remains an open challenge, with potential routes through unitarity methods or reformulations. The work thus presents a promising route to a two-loop BCJ-like framework and deepens connections between CHY/ambitwistor formalisms and standard quantum-field-theory calculations.

Abstract

We propose new formulae for the two-loop n-point D-dimensional integrands of scattering amplitudes in Yang-Mills theory and gravity. The loop integrands are written as a double-forward limit of tree-level trivalent diagrams, and are inferred from the formalism of the two-loop scattering equations. We discuss the relationship between the formulae for non-supersymmetric theories and the Neveu-Schwarz sector of the formulae for maximally supersymmetric theories, which can be derived from ambitwistor strings. An important property of the loop integrands is that they are expressed in a representation that includes linear-type propagators. This representation exhibits a loop-level version of the colour-kinematics duality, which follows directly from tree level via the double-forward limit.

Figures (19)

• Figure 1: The tree-level BCJ master diagrams are half-ladder diagrams with fixed endpoints, which we choose to be legs $1$ and $n$.
• Figure 2: Example at 4 points of how a diagram relates to master diagrams via Jacobi relations. This equation applies to the colour factors of the diagrams. We shall see that it applies also to their kinematic BCJ numerators.
• Figure 3: Sphere with a single node.
• Figure 4: The one-loop BCJ master diagrams are half-ladder diagrams with fixed endpoints, which we choose to be the legs $+$ and $-$ associated to the loop momentum.
• Figure 5: Sphere with two nodes.