Effective Field Theories from Soft Limits

TL;DR

This work reframes scalar EFT construction as a soft-bootstrap problem: by building on-shell tree amplitudes constrained only by Lorentz invariance, factorization, a fixed derivative counting $\rho=m/n$, and a soft limit degree $\sigma$, it classifies theories via the pair $(\rho,\sigma)$. The authors define amplitude ansatze and soft-limit constraints, solve for consistent amplitudes, and show that enhanced soft behavior selects specific Lagrangians, including the Dirac-Born-Infeld (DBI) scalar for $(\rho,\sigma)=(1,2)$ and Galileon theories for $(\rho,\sigma)=(2,2)$ (with truncations to Galileon$_4$ and Galileon$_{4,5}$ emerging). Using LSZ reduction and locality, these soft limits are translated into Noether-current relations that reproduce the corresponding Lagrangians, thereby linking symmetry structures to amplitude constraints. The approach provides a principled route to classify and potentially discover new scalar EFTs from their soft behavior, highlighting the deep connection between infrared soft limits and effective field theory content.

Abstract

We derive scalar effective field theories - Lagrangians, symmetries, and all - from on-shell scattering amplitudes constructed purely from Lorentz invariance, factorization, a fixed power counting order in derivatives, and a fixed order at which amplitudes vanish in the soft limit. These constraints leave free parameters in the amplitude which are the coupling constants of well-known theories: Nambu-Goldstone bosons, Dirac-Born-Infeld scalars, and Galileons. Moreover, soft limits imply conditions on the Noether current which can then be inverted to derive Lagrangians for each theory. We propose a natural classification of all scalar effective field theories according to two numbers which encode the derivative power counting and soft behavior of the corresponding amplitudes. In those cases where there is no consistent amplitude, the corresponding theory does not exist.