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.
Highlights
While much of this work has centered on gauge theory and gravity, another important class of theories — effective field theories (EFTs) — have received substantially less attention, even though they play an important and ubiquitous role in many branches of physics
We systematically explore the space of scalar effective field theories (EFTs) consistent with a Lorentz invariant and local S-matrix
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
Summary
As described in the introduction, scalar EFTs are naturally classified in terms of the set of parameters (ρ, σ, v, d). Eq (2.1) is schematic, since we have suppressed Lorentz and internal indices so at a given order in m, n there are many coupling constants λm,n This restriction still leaves a huge parameter space of viable EFTs. In principle, one can combine interactions of different values of ρ into the same theory. The leading cubic vertex in a derivatively coupled theory of massless scalars can always be eliminated by equations of motion. This is obvious because the only possible non-zero 3pt amplitude of scalars is a constant, corresponding to a cubic scalar potential interaction. We leave the study of multiple ρ theories to future work
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