Ambitwistor formulations of $R^2$ gravity and $(DF)^2$ gauge theories

TL;DR

This work develops CHY-type representations and ambitwistor-string interpretations for $D$-dimensional $R^2$ gravity and the $(DF)^2$ gauge theory, clarifying their relation through color-kinematics duality and the double-copy construction. It introduces a simple CHY formula for the $(DF)^2$ theory, a CHY/ambitwistor formulation for conformal gravity (the $D$-dimensional Berkovits–Witten sector), and a novel $(DF)^2$-photon theory that couples to Einstein gravity. The results reveal how non-unitary higher-derivative theories can be organized in a unitary-looking double-copy framework and show that ambitwistor strings furnish a natural unifying perspective, including connections to Witten's twistor string. These insights pave the way for new KLT-like constructions and a deeper understanding of how higher-derivative gravity and gauge theories fit into the CHY/ambitwistor paradigm.

Abstract

We consider $D$-dimensional amplitudes in $R^2$ gravities (conformal gravity in $D=4$) and in the recently introduced $(DF)^2$ gauge theory, from the perspective of the CHY formulae and ambitwistor string theory. These theories are related through the BCJ double-copy construction, and the $(DF)^2$ gauge theory obeys color-kinematics duality. We work out the worldsheet details of these theories and show that they admit a formulation as integrals on the support of the scattering equations, or alternatively, as ambitwistor string theories. For gravity, this generalizes the work done by Berkovits and Witten on conformal gravity to $D$ dimensions. The ambitwistor is also interpreted as a $D$-dimensional generalization of Witten's twistor string (SYM + conformal supergravity). As part of our ambitwistor investigation, we discover another $(DF)^2$ gauge theory containing a photon that couples to Einstein gravity. This theory can provide an alternative KLT description of Einstein gravity compared to the usual Yang-Mills squared.