Hybrid Goldstone Modes from the Double Copy Bootstrap

We perform a systematic classification of scalar field theories whose amplitudes admit a double copy formulation and identify two building blocks at four-point and 13 at five-point. Using the four-point blocks simultaneously as bootstrap seeds, this naturally leads to a single copy theory that is a gauged nonlinear sigma model. Moreover, its double copy includes a novel theory that can be written in terms of Lovelock invariants of an induced metric, and includes Dirac-Born-Infeld and the special Galileon in specific limits. The amplitudes of these Goldstone modes have two distinct soft behavior regimes, corresponding to a hybrid of nonlinear symmetries.