  Conformal supergravity amplitudes are obtained from the double-copy
construction using gauge-theory amplitudes, and compared to direct calculations
starting from conformal supergravity Lagrangians. We consider several different
theories: minimal ${\cal N}=4$ conformal supergravity, non-minimal ${\cal N}=4$
Berkovits-Witten conformal supergravity, mass-deformed versions of these
theories, as well as supersymmetry truncations thereof. Coupling the theories
to a Yang-Mills sector is also considered. For all cases we give the gravity
Lagrangians that the double copy implicitly generates. The two main results
are: we determine a Lagrangian for the non-minimal Berkovits-Witten theory, and
we uncover the double-copy prescription for the minimal ${\cal N}=4$ conformal
supergravity.

High Energy Physics - Theory