Double copy for Lagrangians at trilinear order

We present a novel double-copy prescription for gauge fields at the Lagrangian level and apply it both to the original double copy and the soft theorem. The Yang-Mills Lagrangian in light-cone gauge is mapped directly to the $mathcal{N}=0$ supergravity Lagrangian in light-cone gauge to trilinear order, and we show that the obtained result is manifestly equivalent to Einstein gravity at tree level up to this order. The application of the double-copy prescription to the soft-collinear effective QCD Lagrangian yields an effective description of an energetic Dirac fermion coupled to the graviton, Kalb-Ramond, and dilaton fields, from which the fermionic gravitational soft and next-to-soft theorems follow.