A Periodic Table of Effective Field Theories

TL;DR

This paper classifies scalar effective field theories based on their properties, identifies the boundary theories like the non-linear sigma model and Dirac-Born-Infeld, and shows constraints on their soft behaviors and interaction structures.

Contribution

It provides a systematic classification of scalar EFTs, proves boundary theories are exceptional, and rules out certain soft behaviors and high valency interactions.

Findings

Exceptional EFTs lie on the boundary of allowed theory space.

EFTs with arbitrarily soft behavior are forbidden.

High valency interactions cannot have enhanced soft behavior.

Abstract

We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix. To do so we define an EFT classification based on four parameters characterizing 1) the number of derivatives per interaction, 2) the soft properties of amplitudes, 3) the leading valency of the interactions, and 4) the spacetime dimension. Carving out the allowed space of EFTs, we prove that exceptional EFTs like the non-linear sigma model, Dirac-Born-Infeld theory, and the special Galileon lie precisely on the boundary of allowed theory space. Using on-shell momentum shifts and recursion relations, we prove that EFTs with arbitrarily soft behavior are forbidden and EFTs with leading valency much greater than the spacetime dimension cannot have enhanced soft behavior. We then enumerate all single scalar EFTs in d<6 and verify that they correspond to known theories in the literature. Our results suggest that the exceptional theories are the natural EFT analogs of gauge theory and gravity because they are one-parameter theories whose interactions are strictly dictated by properties of the S-matrix.