By considering scattering amplitudes in scalar effective field theory, one can categorize different theories according to four numbers: ρ, the number of derivatives per interaction; σ, the soft amplitude's scaling with momenta; v, the number of fields in the leading interaction; d, the spacetime dimension. For a given (ρ,σ,v,d), an ansatze is given for tree level scattering amplitudes which satisfies certain consistency relations (a factorizable combination of Lorentz invariants with the desired pole structure). By demanding S matrix consistency, and an `enhanced' soft behaviour (i.e. that the soft amplitude vanishes faster than naively expected, suggesting some additional asymptotic symmetry), it is possible to restrict the parameter space to a narrow window, which contains the various already known scalar field theories.

Categorizing scalar EFTs by the properties of their scattering amplitudes is far more useful by the usual Lagrangian description, as the latter suffers from integration-by-parts and equations of motion ambiguities even classically. Particularly interesting is that direct enumeration of the `allowed' scalar EFTs with enhanced soft symmetry reproduces all of the known theories (NLSM, DBI, special Galileon, WZW). There are no other scalar fields one could construct with enhanced asymptotic symmetry. Of course, this analysis applies exclusively to massless scalars (else there is no shift symmetry/Adler zero) and on flat spacetime. It is also restricted to tree-level.