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We consider $D$-dimensional amplitudes in $R^2$ gravities (conformal gravityin $D=4$) and in the recently introduced $(DF)^2$ gauge theory, from theperspective of the CHY formulae and ambitwistor string theory. These theoriesare related through the BCJ double-copy construction, and the $(DF)^2$ gaugetheory obeys color-kinematics duality. We work out the worldsheet details ofthese theories and show that they admit a formulation as integrals on thesupport of the scattering equations, or alternatively, as ambitwistor stringtheories. For gravity, this generalizes the work done by Berkovits and Wittenon conformal gravity to $D$ dimensions. The ambitwistor is also interpreted asa $D$-dimensional generalization of Witten's twistor string (SYM + conformalsupergravity). 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 gravitycompared to the usual Yang-Mills squared.

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