Double-soft behavior of the dilaton of spontaneously broken conformal invariance

The Ward identities involving the currents associated to the spontaneously broken scale and special conformal transformations are derived and used to determine, through linear order in the two soft-dilaton momenta, the double-soft behavior of scattering amplitudes involving two soft dilatons and any number of other particles. It turns out that the double-soft behavior is equivalent to performing two single-soft limits one after the other. We confirm the new double-soft theorem perturbatively at tree-level in a $D$-dimensional conformal field theory model, as well as nonperturbatively by using the "gravity dual'' of ${\cal{N}}=4$ super Yang-Mills on the Coulomb branch; i.e. the Dirac-Born-Infeld action on AdS${}_5 \times S^5$.