Gluons and gravitons at one loop from ambitwistor strings

TL;DR

This paper extends the ambitwistor-string and CHY scattering-equations framework to explicit one-loop amplitudes in non-supersymmetric Yang–Mills theory and gravity, revealing a loop-level colour–kinematics duality and a usable double-copy construction. It introduces a novel loop-integrand representation with predominantly linear propagators, derived from forward limits on a nodal Riemann sphere, and provides a practical algorithm to generate all-multiplicity one-loop BCJ master numerators. The authors validate their formulae through gauge-invariance checks, four-point comparisons with known BCJ forms, and dimension-specific tests (all-plus and one-minus in 4D), and they show how gravity amplitudes follow from the double copy of gauge-theory numerators, including a subtraction to obtain pure gravity in four dimensions. Overall, the work demonstrates that loop BCJ duality and the gauge–gravity correspondence extend to non-supersymmetric theories within this representation, offering new computational tools and insights into gravity as the square of gauge theory. The results pave the way for efficient loop calculations, potential unitarity-based approaches, and further exploration of higher-loop extensions within ambitwistor-string theory.

Abstract

We present new and explicit formulae for the one-loop integrands of scattering amplitudes in non-supersymmetric gauge theory and gravity, valid for any number of particles. The results exhibit the colour-kinematics duality in gauge theory and the double-copy relation to gravity, in a form that was recently observed in supersymmetric theories. The new formulae are expressed in a particular representation of the loop integrand, with only one quadratic propagator, which arises naturally from the framework of the loop-level scattering equations. The starting point in our work are the expressions based on the scattering equations that were recently derived from ambitwistor string theory. We turn these expressions into explicit formulae depending only on the loop momentum, the external momenta and the external polarisations. These formulae are valid in any number of spacetime dimensions for pure Yang-Mills theory (gluon) and its natural double copy, NS-NS gravity (graviton, dilaton, B-field), and we also present formulae in four spacetime dimensions for pure gravity (graviton). We perform several tests of our results, such as checking gauge invariance and directly matching our four-particle formulae to previously known expressions. While these tests would be elaborate in a Feynman-type representation of the loop integrand, they become straightforward in the representation we use.

Figures (5)

• Figure 1: The tree-level BCJ master numerators $N(1,\rho(2,...,n-1),n)$, correspond to half-ladder diagrams with legs $1$ and $n$ at opposite endpoints.
• Figure 2: Example at 4 points of how a diagram relates to master diagrams via Jacobi relations. This equation applies to both the colour factors and (by definition) the BCJ numerators associated to the diagrams.
• Figure 3: Diagrammatic depiction: interpretation of the $\mathfrak{I}$ representation of loop integrands as $(n+2)$-particle tree diagrams.
• Figure 4: The one-loop BCJ master numerators $N(1,\rho(2,...,n-1),n)$ correspond to half-ladder diagrams with legs $+$ and $-$ at opposite endpoints.
• Figure 5: Graphic depiction of the Jacobi identity giving rise to the triangle numerator in \ref{['equ:jac_triangle']}.