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We derive positivity bounds on low energy effective field theories whichadmit gapped, analytic, unitary, Lorentz invariant, and possibly non-local UVcompletions, by considering 2 to 2 scatterings of Jaffe fields whoseLehmann-K\"{a}ll\'{e}n spectral density can grow exponentially. Severalproperties of S-matrix, such as analyticity properties, are assumed in ourderivation. Interestingly, we find that some of the positivity bounds obtainedin the literature, such as sub-leading order forward-limit bounds, must besatisfied even when UV completions fall into non-localizable theories inJaffe's language, unless momentum space Wightman functions grow too rapidly athigh energy. Under this restriction on the growth rate, such bounds may provideIR obstructions to analytic, unitary, and Lorentz invariant UV completions.

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