Effective Actions for Domain Wall Dynamics

We introduce a systematic method to derive the effective action for domain walls directly from the scalar field theory that gives rise to their solitonic solutions. The effective action for the Goldstone mode, which characterizes the soliton's position, is shown to consist of the Nambu-Goto action supplemented by higher-order curvature invariants associated to its worldvolume metric. Our approach constrains the corrections to a finite set of Galileon terms, specifying both their functional forms and the procedure to compute their coefficients. We do a collection of tests across various models in $2+1$ and $3+1$ dimensions that confirm the validity of this framework. Additionally, the method is extended to include bound scalar fields living on the worldsheet, along with their couplings to the Goldstone mode. These interactions reveal a universal non-minimal coupling of these scalar fields to the Ricci scalar on the worldsheet. A significant consequence of this coupling is the emergence of a parametric instability, driven by interactions between the bound states and the Goldstone mode.