A Periodic Table of Effective Field Theories

TL;DR

This work maps the space of Lorentz-invariant, local scalar EFTs by constraining their on-shell amplitudes with four parameters (ρ,σ,v,d). It develops an on-shell soft bootstrap and recursion framework to determine which EFTs are allowed, showing exceptional theories (NLSM, DBI, special Galileon) sit on the boundary and saturate soft-limit bounds. The authors prove that arbitrarily enhanced soft behavior is forbidden and that higher-point locality restricts theories to a small set, then enumerate all single-scalar EFTs in d<6, recovering the known exceptional theories and variants (WZW, Galileon). They further extend the analysis to multiple scalars and other kinematic regimes, highlighting deep connections to CHY/BCJ structures and suggesting a unifying amplitude-based perspective on EFTs with potential for discovering new theories. Overall, exceptional EFTs emerge as natural, highly constrained analogs of gauge theory and gravity, fully determined by S-matrix properties.

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.

Figures (4)

• Figure 1: Plot summarizing the allowed parameter space of EFTs. The blue region denotes EFTs whose soft behavior is trivial due to the number of derivatives per interaction. The red region is forbidden by consistency of the S-matrix, as discussed in Sec. \ref{['sec:theory_space']}. The white region denotes EFTs with non-trivial soft behavior, with solid black circles representing known standalone theories. The $d$-dimensional WZW term theory corresponds to $(\rho,\sigma)=(\frac{d-2}{d-1},1)$. The exceptional EFTs all lie on the boundary of allowed theory space and $(\rho,\sigma)=(3,3)$ is forbidden.
• Figure 2: Factorization channel with spurious pole $a_{12}$.
• Figure 3: Plot summarizing the numerical search of EFTs with $v=5,6$. The blue region denotes the same trivial region as in Figure \ref{['fig:EFT_map']}. The red region has no solution numerically. The only two points with solutions are the $d$-dimensional WZW theory, $(\rho,\sigma)=(\frac{v-3}{v-2},1)$, and the Galileon $(\rho,\sigma)=(2,2)$. The label "Sol:{a,b}" denotes the number of solutions in permutation invariant and cyclic invariant amplitudes respectively.
• Figure 4: Plot summarizing the numerical search of EFTs with $v=4$. The blue region denotes the same trivial region as in Figure \ref{['fig:EFT_map']}. The red region has no solution numerically. The label "Sol:{a,b}" denotes the number of solutions in permutation invariant and cyclic invariant amplitudes respectively.