Positivity Bounds for Scalar Theories

TL;DR

The paper addresses how unitarity, analyticity, and crossing symmetry of a UV-complete, Poincaré-invariant scalar theory impose rigorous positivity constraints on the low-energy effective field theory (LEEFT). Using dispersion relations and a pole-subtracted amplitude B(s,t), the authors first establish positivity for no-t-derivatives (B^{(2N,0)}(t) > 0) and then construct a general method to guarantee positivity for any number of nonzero-t derivatives by forming positive combinations Y^{(2N,M)}(t). These results translate into concrete bounds on EFT coefficients a_{nm} and can even bound the mass scale of the next heavy state beyond the EFT (Lambda_th) in certain cases. Overall, the work provides an infinite family of non-forward positivity bounds, sharpening constraints on EFT parameter spaces beyond forward-limit analyses and connecting them to the possible UV completions of scalar theories.

Abstract

Assuming the existence of a local, analytic, unitary UV completion in a Poincaré invariant scalar field theory with a mass gap, we derive an infinite number of positivity requirements using the known properties of the amplitude at and away from the forward scattering limit. These take the form of bounds on combinations of the pole subtracted scattering amplitude and its derivatives. In turn, these positivity requirements act as constraints on the operator coefficients in the low energy effective theory. For certain theories these constraints can be used to place an upper bound on the mass of the next lightest state that must lie beyond the low energy effective theory if such a UV completion is to ever exist.