# Extension of positivity bounds to non-local theories: IR obstructions to Lorentz invariant UV completions

## TL;DR

This paper extends positivity bounds to non-local theories, showing that certain IR constraints can obstruct the existence of Lorentz invariant UV completions with specific growth properties.

## Contribution

It generalizes positivity bounds to non-local theories, revealing IR obstructions to Lorentz invariant UV completions based on spectral growth conditions.

## Key findings

- Positivity bounds apply to non-localizable theories under growth restrictions.
- Certain bounds serve as IR obstructions to UV completions.
- Growth rate of spectral functions determines the applicability of bounds.

## Abstract

We derive positivity bounds on low energy effective field theories which admit gapped, analytic, unitary, Lorentz invariant, and possibly non-local UV completions, by considering 2 to 2 scatterings of Jaffe fields whose Lehmann-K\"{a}ll\'{e}n spectral density can grow exponentially. Several properties of S-matrix, such as analyticity properties, are assumed in our derivation. Interestingly, we find that some of the positivity bounds obtained in the literature, such as sub-leading order forward-limit bounds, must be satisfied even when UV completions fall into non-localizable theories in Jaffe's language, unless momentum space Wightman functions grow too rapidly at high energy. Under this restriction on the growth rate, such bounds may provide IR obstructions to analytic, unitary, and Lorentz invariant UV completions.