  We argue that conformal invariance is a common thread linking several scalar
effective field theories that appear in the double copy and scattering
equations. For a derivatively coupled scalar with a quartic ${\cal O}(p^4)$
vertex, classical conformal invariance dictates an infinite tower of additional
interactions that coincide exactly with Dirac-Born-Infeld theory analytically
continued to spacetime dimension $D=0$. For the case of a quartic ${\cal
O}(p^6)$ vertex, classical conformal invariance constrains the theory to be the
special Galileon in $D=-2$ dimensions. We also verify the conformal invariance
of these theories by showing that their amplitudes are uniquely fixed by the
conformal Ward identities. In these theories, conformal invariance is a much
more stringent constraint than scale invariance.
